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Optimization Models and Algorithms for Joint
Uplink/Downlink UMTS Radio Network Planning
with SIR-based Power ControlAmin Abdel Khalek, Lina Al-Kanj, Zaher Dawy, Senior Member, IEEE, and George Turkiyyah
Abstract UMTS networks should be deployed accordingto cost-effective strategies that optimize a cost objective andsatisfy target quality of service (QoS) requirements. In thispaper, we propose novel algorithms for joint uplink/downlinkUMTS radio planning with the objective of minimizing totalpower consumption in the network. Specifically, we define twocomponents of the radio planning problem: (1) Continuous-based site placement, and (2) Integer-based site selection. Inthe site placement problem, our goal is to find the optimal
locations of UMTS base stations in a certain geographic areawith a given user distribution to minimize the total powerexpenditure such that a satisfactory level of downlink and uplinksignal-to-interference ratio (SIR) is maintained with boundedoutage constraints. We model the problem as a constrainedoptimization problem with SIR-based uplink and downlink powercontrol scheme. An algorithm is proposed and implemented usingpattern search techniques for derivative-free optimization withaugmented Lagrange multiplier estimates to support generalconstraints. In the site selection problem, we aim to select theminimum set of base stations from a fixed set of candidatesites that satisfies quality and outage constraints. We developan efficient elimination algorithm by proposing a method forclassifying base stations that are critical for network coverageand quality of service. Finally, the problem is reformulated to
take care of location constraints whereby the placement of basestations in a subset of the deployment area is not permitted dueto, e.g., private property limitations or electromagnetic radiationconstraints. Experimental results and optimal tradeoff curves arepresented and analyzed for various scenarios.
Index Terms Cellular network planning, network optimiza-tion, network deployment, electromagnetic radiation exposure
I. INTRODUCTION
In UMTS networks, the base station (BS) coverage and
capacity are a function of the user distribution, the signal-
to-interference ratio (SIR) requirements, and the interference
level which is an important coverage-limiting factor. The trans-mit powers of mobile users are power controlled depending
Copyright (c) 2011 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].
This work was supported by a research grant from the National Councilfor Scientific Research (CNRS), Lebanon.
A. Abdel Khalek was with the Department of Electrical and ComputerEngineering, American University of Beirut, Beirut, Lebanon. He is now withthe Department of Electrical and Computer Engineering, The University ofTexas at Austin, TX, USA (email: [email protected]). L. Al-Kanj andZ. Dawy are with the Department of Electrical and Computer Engineering,American University of Beirut, Beirut, Lebanon (email: [email protected],[email protected]). G. Turkiyyah is with the Department of Computer Science,American University of Beirut, Beirut, Lebanon (email: [email protected]).
on their distance from the BS to reduce interference, avoid
the near-far problem and ensure coverage for users close to
the cell edge. Given the high cost of network infrastructure
investments and spectrum licenses, operators should make
informed decisions on network deployment to satisfy perfor-
mance requirements in a cost efficient way. This drives the
need for optimized UMTS-specific planning tools that take
into account the WCDMA air interface characteristics.
UMTS radio network planning involves configuring the
network resources and parameters in a way that guarantees
satisfactory performance for the end-users according to the
following three main quality attributes: coverage, capacity
and quality of service. Radio network planning is conven-
tionally approached as an iterative process which requires
setting target coverage and capacity objectives. The initial
network plan is obtained from geographic data, demographic
data, and propagation models, e.g., [1], [2], [3], and is then
optimized by iterative updates of various network variables.
Several modeling techniques are feasible and can be solved
by mathematical and heuristic optimization algorithms such
as simulated annealing, greedy algorithms, genetic algorithms,linear and non-linear programming, etc.
A. Related Work
In [4], [5], [6], the authors propose discrete optimization
algorithms using randomized greedy procedures and a tabu
search algorithm to plan the process of locating new BSs
considering quality constraints for the uplink which is argued
to be more stringent than the downlink for symmetric traffic. In
[7], the previous work is extended to the downlink for asym-
metric traffic by applying SIR-based power control. Models
spanning both downlink and uplink with power control are
also presented in [8], [9].
In [10], [11], mixed integer linear programming (MILP) is
used for planning cost-efficient radio networks under network
quality constraints. Models based on set covering are used to
obtain lower bounds on the number of required BSs to serve a
given fixed area and an automatic two phase network planning
approach based on successively solving instances of the model
is presented. In [12] and [13], two graph theory based models
are proposed: Maximum independent set (MIS) model and
minimum dominating set (MDS) model to select a satisfactory
subset out of a user-provided set of BS locations, while
ensuring that at least a given percentage of the considered area
is served by the selected BSs. A large set of candidate BS sites
is first determined. In [14], a net-revenue maximization model
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for the selection of BS sites and the calculation of service
capacity is presented. The integer programming model takes
the candidate BS locations and the traffic demand model as
input, and uses a priority branching scheme to achieve a target
optimization gap tolerance.
The majority of the contributions on optimized network
planning focus on locating BSs according to the best trade-
off between network infrastructure costs and service coverage,whereas the electromagnetic (EM) field exposure levels are
rarely considered. However, raising concerns about serious
consequences for human health due to exposure to EM fields
have led to precautionary regulations enforced by public
administrators [15], [16]. The most widely accepted standards
are those developed by the IEEE [17] and the ICNIRP [18].
Due to the increasing concerns about EM pollution in cellular
networks, it should be considered an important metric for
network planning and optimization. The inclusion of EM
radiation in the cellular network planning problem has been
addressed in [19], [20] where two sequential meta-heuristics
were developed to limit the total EM field at selected test
points and combined in a tabu search (TS) and a genetic algo-
rithm (GA). The TS procedure is able to explore the solution
space deeply enough, but it works on partial configurations,
while the GA procedure can manage the complete set of
considered parameters, but is computationally expensive.
Current work in the literature focuses on selecting a minimal
BS set from a larger candidate fixed BS set. The equally-
important continuous version of the problem which involves
finding the optimal locations of these BSs in the network area
was not considered previously in the literature. By combining
the two components of the problem, we can target more
general application scenarios and further adapt and optimize
the network plan.
B. Contributions of the Paper
In this work, the problem of joint uplink/downlink radio
network planning is subdivided into two components. In the
first component, referred to as the site placement problem
throughout the paper, we are interested in finding the optimal
locations of a fixed number of UMTS base stations. In the
second component, referred to as the site selection problem,
we are interested in finding the minimal cardinality set from
a set of base stations with fixed locations. It can be seen that
the site placement problem is a continuous problem because
the problem variables are the physical BS locations, while
the site selection problem is a combinatorial problem because
the problem variables are the binary selection variables for
each of the BSs. We solve the continuous problem of finding
the optimal locations of base stations that minimize total
power expenditure in the network using robust pattern search
algorithms for derivative-free optimization with non-smooth
objectives. We define QoS targets and maintain bounded
outage levels on a network-wide and per BS basis for both
uplink and downlink channels. Next, we formulate and propose
an algorithm to solve the integer problem of selecting the
smallest set of BSs from a fixed set of potential sites such
that the cost function is minimized and the QoS requirements
are satisfied.
It is important to note that these two problems and their
solution approaches are distinct, however, we propose that the
two problems be combined into a hybrid integer/continuous
algorithm involving successive site selection/site placement
until both components converge, i.e., until no relocation or
elimination is feasible. This allows the operator to find the
minimal set of BSs needed for satisfactory coverage and the
optimized deployment strategy in the network according to theuser distribution. An additional novel component of our work
is the development of a framework for radio network planning
with location constraints. Such constraints arise frequently in
practice due to private property limitations or electromagnetic
radiation constraints. We formulate the problem of radio
network planning with location constraints as an optimization
problem. As part of the problem solution, we use Lagrangian
multiplier estimates and penalty parameters to construct and
solve a sequence of augmented Lagrangian subproblems based
on the Augmented Lagrangian Pattern Search (ALPS) algo-
rithm. Finally, we provide optimal tradeoff curves under dif-
ferent user distributions and we demonstrate the effectiveness
of the proposed schemes compared to conventional clustering.
C. Paper Organization
The rest of the paper is organized as follows: Section II
describes the system model. Section III presents the math-
ematical formulation for the site selection and site place-
ment optimization problems. Section IV presents the strategy
and algorithms used to solve the optimization problems. In
Section V, we report results for different user distributions
and voice/data traffic combinations and we provide optimal
tradeoff curves for the network configuration. In Section
VI, we extend the optimization algorithms to scenarios with
location constraints and/or EM radiation restrictions. Finally,Section VII provides concluding remarks.
II . SYSTEM MODEL
The user distribution model is assumed to be snapshot-
based. A snapshot represents a set of users (or test points)
using the physical channel at a given instant of time. For a
given distribution of currently active users or connections, we
aim to find a network plan that guarantees minimal power
consumption in the network, and satisfies coverage and quality
of service requirements in a cost-effective manner.
In UMTS, users rely on channelization and scrambling
codes in order to differentiate their own signal and combatthe effect of multipath and multiuser interference. To guarantee
the required QoS level, a target minimum SIR value should be
maintained for all active connections. The downlink and uplink
SIR expressions for user k can be expressed as follows:
SIRdk = SFPdreceived,k
2 + dIdin,k + Idout,k
(1)
SIRuk = SFPureceived,k
2 + uIuin,k + Iuout,k
(2)
where Pdreceived,k is the downlink received power at mobilestation (MS) k, Pureceived,k is the uplink received power at
the BS from MS k, Idin,k and Idout,k represent the downlink
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SF + SIRdk
d
SIRdk
gb(k),kP
db(k),k
dgb(k),k
jCb(k)
Pb(k),j
N
i=1,i=b(k)
gi,k
lCi
Pdi,l
= 2 (3)
SF + SIRuk
u
SIRuk
gb(k),kP
uk
u
jCb(k)
gb(k),jPuj
N
i=1,i=b(k)
lCi
gi,lPdl
= 2 (4)
intracell and intercell interference affecting MS k, respectively,Iuin,k and I
uout,k represent the uplink intracell and intercell
interference affecting MS k, respectively, SF is the spreadingfactor, 2 is the thermal noise power, and is the orthogonalityfactor (0.4 d 0.9 in the downlink and u = 1 inthe uplink [21]). Since we need to satisfy the SIR constraint
for all users, we express the received power and interference
components in (1) and (2) more explicitly for user k asfollows:
Pdreceived,k = gb(k),kPdb(k),k; P
ureceived,k = gb(k),kP
uk (5)
Idin,k = gb(k),k Pdb(k)Pdb(k),k ; Iuin,k = jCb(k),j=k
gb(k),jPuj (6)
Idout,k =
Ni=1,i=b(k)
gi,kPdi ; I
uout,k =
Ni=1,i=b(k)
jCi
gi,jPuj (7)
where N is the number of BSs, Pdi is the total transmit powerof BS i, Pdi,k is the power allocated by BS i to MS k (subjectto k being covered by BS i), gi,k is an estimate of the pathlossbetween MS k and BS i calculated according to Cost 231-Hatamodel [1], and b(k) is defined as the BS serving MS k andcalculated by constructing the Voronoi tessellations associated
with the BS locations. Consequently, Pdb(k),k is the transmitpower allocated to MS k by its serving BS, Pb(k) is the totaltransmit power of the BS serving user k and gb(k),k is anestimate of the pathloss between MS k and its serving BS.The notation j Ci means all MSs {j} covered by cell i.Consequently, j Cb(k) means all MSs covered by the sameBS as k. The pathloss coefficients gi,k can be written strictly interms of the BS and MS locations in addition to some constants
as gi,k =1keq
(di,k), di,k =
(xi uk)2 + (yi vk)2
where (xi, yi) are the BS locations and (uk, vk) are the fixedMS locations, with keq and chosen according to Cost-231Hata Model. In UMTS radio network planning, shadowing and
fading are compensated for via link budget margins [1], [21].
Based on the derivation above, (1) can be rewritten in terms
of the powers Pdi,k allocated to MSs which are the downlinkstate variables as shown in (3). Similarly, (2) can be rewritten
in terms of the MS transmit powers Puk which are the uplinkstate variables as shown in (4) where SIRdk and SIR
uk are the
target SIRs to achieve the required QoS for the downlink andthe uplink respectively. Equations (3) and (4) can be expressed
in matrix format as follows:GdUU
PdU1
=
2U1
(8)
[Gu]UU [ Pu ]U1 =
2U1
(9)
where Gd and Gu are square matrices with size U U, Pd is acolumn vector with size U 1 corresponding to the downlink
user powers Pdb(k),k, Pu
is a column vector with size U 1
corresponding to the uplink user powers Puk , 2 is the thermalnoise power column vector with size U 1, and U is the totalnumber of active users in the network. The kth row of Gd andGu corresponds to user k with each row representing one ofthe U equations, and the columns can be separated into blockscorresponding to the BSs according to the number of users in
each BS. The first term of (3) and (4) appears in the diagonal
ofGd and Gu, respectively, the second term represents intracell
interference and appears in the block corresponding to the BS
serving user k, the third term represents intercell interferenceand appears in the blocks corresponding to all BSs except the
serving BS. Solving the power assignment problem with SIR-
based power control reduces to solving this set of equationswhich adjusts the transmit powers in order to to meet the target
SIRs [1], [2].
III. OPTIMIZATION PROBLEM FORMULATION
In this section, we present a formulation for the joint
uplink/downlink radio planning problems. The site placement
problem is formulated as a continuous optimization problem
and the site selection problem is formulated as an integer opti-
mization problem, and the objective, variables, and constraints
for each of the problems are defined.
A. The Joint Uplink/Downlink Site Placement Problem
The input to the site placement problem is: (1) The area
of interest, (2) the fixed set of MS locations modeling the
typical distribution of active users in the area, (3) the fixed
number of BSs, and (4) the initial locations of the BSs. The
site placement algorithm optimizes the initial locations of the
BSs to minimize the target objective while satisfying the QoS
requirements. The output is the set of optimal locations of
BSs. The objective of joint uplink/downlink site placement is
defined as a weighted linear combination of two objectives:
1) Minimize the downlink power expenditure: The total
downlink power expenditure is the sum of powers al-
located to each user by its serving BS, expressed asUk=1 P
db(k),k. In addition to the inherent benefits of
minimizing power consumption, we will demonstrate
that this reduces the variance of BS powers, thus pro-
viding a solution that balances the power load according
to the predicted user distribution.
2) Minimize the uplink outage: The uplink channel is
limited by the power capabilities of the MSs, which is
by far less than that of the BSs. Thus, it is important
to ensure that uplink transmissions can still reach the
BSs at a reasonable power subject to their handset limi-
tations while satisfying the minimum SIR threshold for
acceptable performance. We formulate this component
asU
k=1 (Puk Pumax)+ where ()+ = max(, 0), Puk is
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the uplink power associated with MS k, and Pumax is themaximum allowed MS power.
Effectively, each MS requiring power in excess of Pumaxwill be considered in the second objective. If all MSs do not
exceed the threshold, this will be a vector of zeros, essentially
disregarded from the objective. We do not consider the number
of MSs that are in outage directly in the objective function to
maintain coherence in the units (Watts) and avoid non-smoothstep transitions in the function. However, this approach has the
same potential in minimizing the number of MSs in outage
since the algorithm will attempt to find the BS configuration
that satisfies (Puk Pumax)
+ = 0 k, if such configurationexists.
In the site placement problem, N is fixed, that is, weare not interested in reducing the initial number of BSs but
in optimizing their locations. Thus, the decision variables
are the locations of the BSs (xi, yi), i = 1, , N thatminimize the cost function. Since the objective function is
expressed in terms of the downlink powers assigned by the
BSs to their MSs, we consider the powers assigned by theBSs (Pi,k), i = 1, , N , k = 1, , Mi as state variables.Note that
Ni=1 Mi = U where Mi is the number of MSs
served by BS i. Thus, the total number of variables in theproblem is U + 2N where U N. The power assignmentvariables relate to the decision variables through the SIR-based
power control mechanism as shown in (8) and (9). Since the
matrices Gd and Gu consist of the pathloss coefficients gi,k,it is worth mentioning that it can be written strictly in terms
of the BS and MS locations in addition to some constants
according to the equations gi,k =1keq
(di,k), di,k =
(xi uk)2 + (yi vk)2. Thus, the power assignments for a
given setting of the location variables can be found by solvingthe U set of equations, an operation that costs O(U3).
Two important observations concerning Gd and Gu are
worth pointing out: First, they are strongly diagonally-
dominant. For example, the average value of the diagonal
terms is at least 103 times the average value of the non-diagonal terms with a spreading factor of 128 since the
spreading gain appears only in the diagonal terms and the
non-diagonal terms correspond to interference components.
This fact can be exploited by using fast solvers suitable for
diagonally-dominant matrices. Second, if the SIR requirement
for the kth user cannot be satisfied due to high interferenceat its location, the solution of the corresponding equation will
yield a negative power, which can be used for spotting outages
in the network.
The outage conditions are defined network-wide and for
each BS. The network-wide outage conditions ensure that the
total number of users in outage is less than networkU and theBS outage conditions ensure that the number of users in outage
for every BS i is less than BSMi where network and BS aredesign parameters that satisfy 0 network BS 1.
The problem of joint uplink/downlink site placement is for-
mulated as an optimization problem as in (10)-(20) where (10)
represents the weighted objective and is a constant whichdetermines the relative weight of each of the two components,
(11) represents the matrix-form downlink SIR constraint for all
MSs, (12) represents the matrix-form uplink SIR constraint for
all MSs, (13) is the downlink network outage condition, (14)
is the uplink network outage condition, (15) is the downlink
BS outage condition for every BS, (16) is the uplink BS
outage condition for every BS, (17) is the pathloss according to
Cost 231-Hata model, (18) is the distance-based assignment of
MSs to BSs obtained by constructing the Voronoi tessellation
corresponding to the BS locations, (19) is the maximum BSpower constraint, and (20) is the area of operation constraint.
Note that MS k may be in outage either because it requires atransmission power Puk higher than P
umax to reach the closest
BS, or because the transmit power Puk , although lower thanPumax, causes high interference at another MS so that the SIRconstraints cannot be satisfied. Equation (10) eliminates outage
due to the former case while (13)-(16) eliminate outage due
to the latter case.
minx,y
Uk=1
Pdb(k),k +
Uk=1
(Puk Pumax)
+ (10)
s.t.GdUU
PdU1
=
2U1
(11)
[Gu]UU [ Pu ]U1 =
2U1
(12)
Uk=1
Pdk,b(k)
|Pdk,b(k)|
+< network U (13)
Uk=1
Puk|Puk |
+< network U (14)
kCi
Pdk,b(k)
|Pdk,b(k)|
+< BS Mi (15)
kCi
P
uk
|Puk |
+< BS Mi (16)
gi,k =1
keqdi,k ; di,k=
(xi uk)2 + (yi vk)2(17)
b(k) = argmini di,k (18)jCi
Pi,j Pdmax (19)
xmin xi xmax, ymin yi ymax (20)
i = 1, , N; k = 1, , U (21)
B. The Joint Uplink/Downlink Site Selection Problem
The input to the site selection problem is: (1) The area of
interest, (2) the fixed set of MS locations modeling the typical
distribution of active users in the area, (3) the candidate set of
BSs with fixed locations. The site selection algorithm selects
the minimal cardinality set of BSs that satisfies the coverage
and SIR requirements. The output is the subset of candidates
corresponding to the selected BSs. The main difference from
the site placement problem is that the decision variables are
not the BS locations, instead, these locations are fixed, and
the decision variables are booleans ci where ci = 1 if BS i isto be used in the optimal network configuration, and ci = 0otherwise. The objective of the problem is to minimize the
number of selected BSs, that is,N
0i=1 ci where N0 is the
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size of the candidate set of BSs. The problem can thus be
formulated as a nested optimization problem as shown in (22)-
(24). The inner problem finds the set of BSs that minimizes
the cost function and the outer problem finds the minimal
cardinality set of such BSs. We will later show that using the
minimization of the total power expenditure as the criterion for
BS removal results in better solutions than a removal based
purely on the feasibility of BS elimination (See Section V-B).Note that the constraints (11)-(20) are only applied to the set
of active BSs satisfying ci = 1.
minc
N0i=1
ci (22)
s.t.
minU
k=1
cb(k) Pdk,b(k) +
Uk=1
cb(k) (Puk P
umax)
+(23)
subject to
(11), (12), (13), (14), (15), (16), (17), (18), (19), (20)
i = 1, , N0; k = 1, , U (24)
IV. OPTIMIZATION STRATEGIES AND ALGORITHMS
Solving the initial site placement optimization problem
involves nonlinear equality constraints: (11), (12), (17), and
(18), nonlinear inequality constraints: (13)-(16) and (19), and
linear bound constraints: (20). Given the current BS locations
x and y, we use (18) to obtain the assignments of MSs to
their closest BS, and (17) to obtain the pathloss between each
MS and its serving BS. Finally, solving the SIR equations
(11) and (12), we obtain the power allocated to each MS on
the uplink and downlink to satisfy QoS requirements. Thus,
the four sets of equality constraints are implicitly includedin objective function evaluations because they determine the
power allocation scheme throughout the network. The bound
constraint representing the area of operation is a simple linear
constraint that is taken care of by the algorithm by choosing
appropriate steps that do not violate that constraint. To solve
the site placement problem, we propose a pattern search algo-
rithm based on Mesh Adaptive Direct Search (MADS). Addi-
tionally, to ensure satisfying the BS power limit and the outage
constraints, we will describe how to extend the algorithm to
include any general nonlinear inequality constraints using the
Augmented Lagrangian Pattern Search (ALPS) method. After
presenting the algorithm for the site placement problem, we
will present the solution for the nested optimization problem
of site selection based on successive elimination of BSs.
A. Algorithm for Site Placement with Uplink and Downlink
QoS Guarantees
The MADS class of derivative-free algorithms is effective
for practical nonlinear optimization problems with nonsmooth
objective functions where the computation of derivatives is
either not possible, or it is not sufficiently representative
of the variability of the function around a point due to
the roughness of the objective function. MADS has a well
developed convergence theory based on the Clarke calculus
and Rockafellers notion of a hypertangent cone [22],[23].
To illustrate why the MADS algorithm is suitable for radio
planning problems, we present a sample pattern of the change
of the objective function value with respect to a change in
one of the variables (equivalent to moving one of the BSs
along a coordinate direction). Figure 1 shows that the objective
function has a lot of discontinuities and non-smooth changes
which are explained by the changes in user assignments due
to the shift in the Voronoi diagram. As new users are handed-over to a new BS, they become on the boundary of that BS,
requiring higher power to achieve the target SIR, thus causing
the instant rise in power allocations, and correspondingly,
in the objective function value. Note that the change in
one MS-BS assignment does not affect only that user due
to the changes in intercell and intracell interference effects
experienced by other users, translated in the change of the
structure and content of the pathloss matrices Gd and Gu and
the solutions to the set of equations in (8) and (9). Although
the total power expenditure increases if the BS is moved any
distance larger than 20 meters, a gradient computed using finite
difference method or adjoint method is misleading because
it will imply that the objective function is decreasing which
is only valid 20 meters away from the BS. These kinds of
peculiarities make it impractical to follow a line search method
in solving the problem, which is why we advocate the use of
derivative-free optimization algorithms suitable for the non-
smooth objectives of the radio network planning problem.
0 100 200 300 400 500180
181
182
183
184
185
186
187
188
Movement Distance (m)
TotalPowerExpen
ditureinthenetwork(W)
Fig. 1. Pattern of change of objective function while moving a base stationalong a coordinate direction.
The MADS algorithm operates by performing polling and
searching on a set of mesh points around the current location
of the decision variables. The mesh points are constructed by
building a pattern which is a basis set of vectors that the
algorithm uses to determine which points to search at each
iteration. The most common basis set is the 2n basis set wheren is the number of decision variables. In our problem, thenumber of decision variables is n = 2N, thus the basis set willcontain 4N vectors each of size 2N. The vectors are definedas follows: v1 = [1 0 0], v2 = [0 1 0], , v2N =[0 0 1], v2N+1 = [1 0 0], v2N+2 = [0 1 0], , v4N = [0 0 1].
At each iteration, the pattern search polls the points in the
current mesh for a point that improves the objective function
by computing the objective function at the mesh points to
check if there is one whose function value is less than the
function value at the current point. The mesh points are found
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Algorithm 1 Proposed solution for the joint uplink/downlink
site placement problem.
Given N ; U ; {xi = xinitial,i}2Ni=1 ; {uk = ufixed,k}
2Uk=1 ; =
0 ; th ; ; ; keq ; 2 ; d ; u ; SF ; {SIRdk}
Uk=1 ; {SIR
uk}
Uk=1.
Construct the 2n basis set of vectors v that the algorithm uses to determinewhich points to search at each iteration where n is the number of decisionvariables (2n = 4N)v1 = [1 0 ... 0], v2 = [0 1 ... 0], . .. ,v2N = [0 0 ... 1]v2N+1 = [1 0 ... 0], v2N+2 = [0 1 ... 0],...,v4N = [0 0 ... 1].
while > th{While the mesh size is higher than the convergencethreshold} do
for m = 1 : 4N{For all possible patterns of movement} doxtemp = x + m vpStep 1. Construct the Voronoi tessellation corresponding to the basestation locations {xm} and find the distance between each (BS ,MS) pairStep 2. Construct the pathloss matrices Gd and Gu based on theestimated pathloss for each (BS , MS) pair in the uplink and downlinkStep 3. Solve the set of equations:
Gd
Pd
=
2
and
[Gu] [ Pu ] =
2
to find the downlink power that should beallocated to each MS and the uplink MS power to achieve SIR-basedpower control for all users.Step 4. Calculate the objective function at this mesh point
xm: fm =U
k=1
Pdk,b(k) + U
k=1
(Puk Pumax)
+
end forif minm fm < f{Is there a mesh point with smaller objectivefunction} then
x = x+vm{Successful Poll: Advance to the point that minimizesthe objective and expand mesh size} = (Expansion Factor).
else = (Contraction Factor){Unsuccessful Poll: reduce meshsize}.
end ifend while.
by multiplying each pattern vector vp by a scalar m to gen-erate a set of direction vectors and adding the direction vector
to the current point found at the previous step. The number of
mesh points 4N is simply due to the four coordinate directionsof polling for each BS. These points can be expressed as
xp = x + mvp, p = 1, , 4N where x is a vector ofsize 2N corresponding to the current locations of the N BSs,m is the mesh size at iteration m, and xp is the polled meshpoint. Initially, the mesh size is specified based on the scale of
the problem, and is expanded and contracted during execution
of the optimization algorithm. In this way, the algorithm finds
a sequence of points, x0, x1, x2, that approach an optimalpoint. The convergence criteria are satisfied when the mesh
size is smaller than the mesh tolerance such that minimizing
the objective function further would require moving any BS
a distance smaller than this mesh tolerance threshold. This
polling technique is effectively equivalent to moving the BSs
in their neighborhood according to the mesh size at a given
iteration. It is important to note that at each polled point, the
objective function is calculated by reconstructing the Voronoi
tessellation, building the pathloss matrices Gd and Gu, solving
the set of equations corresponding to SIR-based power control,
and computing the objective function from Pd and Pu. A
complete algorithmic description is presented in Algorithm 1.
As mentioned previously, the algorithm described above
does not account for the nonlinear inequality constraints in the
problem. To include the maximum BS power and the outage
constraints, we use the ALPS algorithm which is a robust
extension of pattern search algorithms for general constraints.
Algorithm 2 Proposed solution for the joint uplink/downlink
site selection problem.
Given N = N0 ; U ; F = ; I = ; {xi = xinitial,i}N0i=1 ; {yi =
yinitial,i}N0i=1 ; ; ; keq ;
2 ; d ; u ; {uk =ufixed,k}
Uk=1 ; {vk = vfixed,k}
Uk=1 ; SF ; {SIR
dk}
Uk=1 ; {SIR
uk}
Uk=1.
while true dofor i = 1 : N{Try eliminating base station i} do
xtemp = {x1, , xi1, xi+1, , xN}ytemp = {y1, , yi1, yi+1, , yN}Step 1. Construct the Voronoi tessellation corresponding to the basestation locations {xtemp, ytemp} and find the distance between each(BS , MS) pairStep 2. Construct the pathloss matrices Gd and Gu based on theestimated pathloss for each (BS , MS) pairStep 3. Solve the set of equations:
Gd
Pd
=
2
and
[Gu] [ Pu ] =
2
to find uplink and downlink power allocationachieving SIR-based power control for all users.Step 4.if Pd and Pu do not satisfy (13)-(16) {If eliminating the base stationcauses significant DL or UL coverage loss} then
i I {Place BS i in the infeasible set}else
i F {Place BS i in the feasible set}Calculate the objective function at (xtemp,ytemp): fi =U
k=1 Pdk,b(k) +
Uk=1 (P
uk
Pumax)+
end ifend forif F = {If some base station can be eliminated} then
e = argminifix = {x1, , xe1, xe+1, , xN}y = {y1, , ye1, ye+1, , yN}N = N 1
elsebreak{Converged: No BS can be eliminated without jeopardizingcoverage}
end ifend while
The algorithm operates by formulating a subproblem obtained
by combining the objective function and nonlinear constraint
functions using Lagrange multiplier estimates and penalty
parameters. A sequence of such optimization problems are
approximately minimized using a pattern search algorithm and
the convergence to the optimal solution is guaranteed (See
Section VI-B for more details).
B. Algorithm for Site Selection with Uplink and Downlink QoS
Guarantees
The site selection problem is a nested optimization problem
with the outer problem minimizing the number of BSs and
the inner problem selecting the set of BSs that minimize the
weighted cost function. The approach for solving the problem
is based on successive elimination of BSs one at a time. Given
a set of fixed BSs S and a user distribution model, at eachelimination step, we can divide the set of BSs S into twodistinct subsets: F and I where F is the feasible set (i.e.,if BS i F, it can be safely eliminated from the set)and I is the infeasible set (i.e., if BS i I, it cannotbe eliminated from the set without jeopardizing coverage to
a significant fraction of users as defined by network andBS) and F I = S. Formally, BS i F if for the set{BS1, , BSi1, BSi+1, , BSN}, a solution for the2U equations in (8) and (9) such that (13)-(16) are satisfied.Out of the feasible set, we eliminate the BS that minimizes
the weighted objective after elimination in accordance with
our initial cost function. This whole operation is repeated until
at some point, the elimination testing phase yields an empty
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TABLE I. SIMULATION PARAMETERS
Parameter Value Description Parameter Value Description
U 1000 Number of active users xmax 10 Km Area dimensionskeq 2.75 10
15 Pathloss parameters ymax 10 Km 3.52 Eb/I0 (DL) 7 dB QoS parameters
Pdmax 30 W Maximum BS power Eb/I0 (UL) 5 dBPumax 1 W Maximum MS power network 0.05
SF 128 Service characteristics BS 0.05Rdvoice 12.2 Kbps
d 0.4 Orthogonality factors
Ruvoice 12.2 Kbps u 1
Rddata 64 Kbps 0.5 Multiobjective weightRudata 32 Kbps
2 2 1014 W Noise power
0 2 Km 4 Km 6 Km 8 Km 10 Km0
2 Km
4 Km
6 Km
8 Km
10 Km
0 2 Km 4 Km 6 Km 8 Km 10 Km0
2 Km
4 Km
6 Km
8 Km
10 Km
0 2 Km 4 Km 6 Km 8 Km 10 Km0
2 Km
4 Km
6 Km
8 Km
10 Km
Fig. 2. Base station distribution obtained for different user distributions with combined site selection and site placement; Left: Uniform user distribution,Middle: Gaussian user distribution, Right: Four Gaussian hot spots. In the figures, stars denote base stations, dots denote mobile stations, and the boundarylines denote Voronoi regions.
feasible set. It is worth noting that testing feasibility requires
solving the set of 2U equations in (8) and (9) for each BS. Acomplete algorithmic description is presented in Algorithm 2.
In order to provide a general optimization framework for
UMTS radio network planning, we combine the two opti-
mization problems of site selection and site placement to
find the minimal set of BSs to cover the area and their
optimal locations subject to the user distribution. Thus, the
two algorithms can be successively executed as subproblems
until the convergence criteria for both are satisfied.
V. RESULTS AND INTERPRETATION
This section presents results and analysis for a 10 Km x10 Km area with U = 1000 active users having the sametarget per-bit signal-to-interference ratio Eb/I0 (DL) = 7dB and Eb/I0 (UL) = 5 dB. Initially, we assume all MSsare operating a voice service. In Section V-D, we present
results for network scenarios with two service classes (voice
and data). For radio propagation, we consider the COST-231
Hata model for metropolitan areas [1] and we compensate
for shadowing, fading, and antenna losses by adding a 16 dB
margin to the link budget. This resulting total loss between BS
i to MS k can be approximated to within less than 0.1 dB in allregions of interest by gi,k =
1keq
di,k where keq = 2.75 1015
and = 3.52. The simulation parameters are summarized inTable I.
Each BS is assumed to be equipped with an omni-directional
antenna that is placed at the cell center. We point out that the
algorithms developed in this paper make no assumptions about
the type or directionality of the antennas. In fact, they can be
easily applied to a sectorized network by accounting for the
directional antenna gains and constructing the Voronoi regions
on a sector by sector basis. In this section, we assume that both
BSs and MSs are equipped with omni-directional antennas.
A. Optimal Base Station Configurations for Different User
Distributions
We present sample results of optimal BS deployments for
three different distributions by executing Algorithm 1 and
Algorithm 2 consecutively to combine site selection and site
placement and find the minimal set of BSs and their optimal
locations to cover the given area. For each of the three
distributions, we start initially with N0 = 100 randomlylocated BSs. N0 is chosen to be large enough to provide aninitial configuration that satisfies outage and SIR requirements.
Figure 2(a) shows the optimal BS distribution for a uniform
user distribution with 1000 active users distributed uniformly
over the entire area. Results show that 64% of the BSs were
eliminated, reaching 36 base stations with an average BS
power of 17.1 W and standard deviation of BS powers of1.26 W. Figure 2(b) shows the optimal BS distribution for aGaussian user distribution modeling a hot spot with maximum
user density at the center and decreasing gradually towards
the area boundary. Results show that 61% of the BSs were
eliminated, reaching 39 BSs with an average BS power of
13.1 W and standard deviation of BS powers of 4.34 W.Finally, Figure 2(c) solves the problem for four Gaussian
user distributions modeling several hot spots with 250 active
users each. After execution, 55% of the BSs were eliminated,
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reaching 45 BSs with an average BS power of 11.6 W and astandard deviation of 3.39 W.
B. Analysis of Site Selection and Site Placement Algorithms
Figure 3 shows the average BS power during execution
of the site selection algorithm for different inner objectives
with a Gaussian user distribution and N0 = 100 BSs. The
outer objective is to minimize the number of BSs and withthe inner objective to either (1) minimize sum of BS powers
or (2) eliminate first feasible BS. In the first approach, we
eliminate the one that among other BSs minimizes total power
expenditure as described in Section III-B. This essentially
provides a solution that minimizes average BS power. In
the second approach, while executing the outer problem, we
greedily eliminate the first possible BS that does not cause
significant coverage loss as defined in (13)-(16). Obviously,
the average BS power will increase under both approaches
since BSs are getting eliminated, however, what the algorithm
optimizes is the rate of this increase. The most noticeable
difference is that 46 BSs were eliminated with the first inner
objective while only 20 BSs were eliminated with the greedy
approach. This demonstrates the impact of the elimination
criteria on the rate of increase of the average BS power, and
thus on the effectiveness of the selection algorithm.
0 5 10 15 20 25 30 35 40 452
3
4
5
6
7
8
Number of Eliminated Base Stations N0 N
Average
BSPower
1 N
U k
=1
Pd k
,b(k)
Inner objective: Min sum of BS powersInner objective: First feasible BS
26 more base stations eliminated
Fig. 3. Improvement in the radio network plan due to the utilization of thenested objective in the site selection algorithm.
Another important observation is that minimizing the total
power expenditure implicitly reduces the variance of BS
powers by converging to a solution that distributes the power
load nearly equally among BSs. Figure 4(a) shows the average
BS power during the execution of the site placement algorithm
with a Gaussian user distribution for two different objective
functions: (1) Minimize sum of BS powers, (2) Minimize
variance of BS powers. The two objectives converge almost to
the same average BS power suggesting that opting for equal
BS powers does not greedily increase these powers to achieve
equality, instead, the network configuration converges to a low
power solution due to the SIR-based power control mechanism
utilized in the problem. Figure 4(b) shows a similar scenario
for the site selection algorithm with N0 = 100 initial BSs.We present a case study to analyze the effect of the initial
number of base stations N0 on the solution. We consider auniform user distribution with U = 200 active users. Initially,
the BSs are located randomly in the network. We execute
the site placement and site selection algorithms successively
as subproblems until the convergence criteria for both are
satisfied. This corresponds to the case where eliminating any
BS causes the quality of service constraints to be violated and
moving any BS a distance larger than the mesh size threshold
will only increase the objective. The resulting network plan
size, average transmit power per BS, global iterations, and total
computation time are shown in Table II for N0 = 100, 60, 40,and 25. Global iterations represents the number of times thesite selection and site placement algorithms are sequentially
executed until the aforementioned convergence criteria are
satisfied.
TABLE II. EFFECT OF N0 ON THE SOLUTION QUALITY
N0 Network Average BS Global Total comp.size (BSs) Tx power (W) iterations time (sec)
100 17 21.32 5 471.15
60 18 16.29 3 203.63
40 20 12.44 2 172.30
25 23 9.51 1 99.80
The network plan size is the main metric for judging the
quality of the solution. It can be observed from the results
that larger N0 generates a finer solution since there are moredegrees of freedom in selecting the candidate sites, thus, more
critical site locations can be picked during the optimization.
Consequently, the solution is only slightly dependent on the
initial locations of BSs. On the other hand, a smaller N0 ismore likely to provide a locally optimal solution. Obviously,
a solution with lower number of BSs would require higher
average transmit power per BS that satisfies the BS maximum
transmit power constraint. Additionally, a larger N0 requires
more global iterations and more computation time.
C. Radio Network Planning Tradeoffs
Combining site selection and site placement allows the op-
erator to find the minimal set of BSs to cover the network area
and their optimal locations. Since eliminating BSs increases
the average power per BS, we can think of the problem
differently by trading-off the number of BSs and the average
BS power. The solid lines of Figure 5 show the set of pareto-
optimal points that provide this tradeoff between the minimum
average BS power achieved and the number of deployed BSs.
The pareto sets are generated for a uniform user distribution in
Figure 5(a) and for a Gaussian user distribution in Figure 5(b).
Generating these sets is performed by running the site selection
algorithm to select a target number of BSs Nmin from aset of 100 initial BSs. The site placement algorithm is then
executed for these Nmin BSs to find their optimal locationsthat minimize total power expenditure.
In an attempt to validate our algorithm and demonstrate its
effectiveness, we compare these results with off-the-shelf hi-
erarchical clustering algorithms. Hierarchical clustering treats
each data point initially as a single cluster, and then succes-
sively merges clusters according to the linkage criteria. In
this problem, the data points are the MSs and each cluster
corresponds to a single BS serving a set of MSs. In complete
linkage hierarchical clustering, also called furthest neighbor,
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0 5 10 15 20 25 30 352
3
4
5
6
7
8
9
10
Iteration
A
verageBSPower
1 N
U k
=1
Pd k
,b(k)
Objective: Min variance o f BS Powers
Objective: Min sum of BS Powers
0 5 10 15 20 25 30 35 40 452
3
4
5
6
7
8
9
10
Number of Eliminated Base Stations N0 N
A
verageBSPower
1 N
U k
=1
Pd k
,b(k)
Inner objective: Min variance of BS powersInner objective: Min sum of BS powers
Fig. 4. Comparison of different cost functions according to the achieved average BS power, Left: Site placement algorithm, Right: Site selection algorithm.
30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Target number of Base Stations Nmin
AverageBSp
ower
1 N
U k
=1
Pk
,b(k)
Site Selection and Site PlacementAverage Linkage ClusteringComplete Linkage Clustering
Operating point which minimizes number
of BSs for a uniform user distribution
30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Target number of Base Stations Nmin
AverageBSp
ower
1 N
U k
=1
Pk
,b(k)
Site Selection and Site PlacementAverage Linkage ClusteringComplete Linkage Clustering
Operating point which minimizes number
of BSs for a Gaussian user distribution
Fig. 5. The optimal tradeoff set between the average BS power and the number of deployed base stations is shown by the solid lines. The dotted lines providecomparison with average linkage and complete linkage clustering algorithms. Left: uniform user distribution, Right: Gaussian user distribution.
the two clusters whose merger has the smallest maximum
pairwise distance are merged. In single linkage clustering, also
called nearest neighbor, the two clusters with the smallestminimum pairwise distance are merged. Finally, in average
linkage clustering, the two clusters with the smallest average
of pairwise distances (maximum cohesion) are merged.
For the considered problem, single linkage clustering is not
a suitable approach due to its tendency to form long chains
that cannot model BS coverage. Thus, we consider the other
two linkage criteria. For a given number of clusters (i.e.,
BSs) Nmin, each of the two criteria is applied to constructthe sets of MS clusters, and the BS location for each cluster
is defined as the centroid of the set of cluster data points.
The power allocated to each user is determined by solving the
equations Gd Pd = 2 and [Gu] [ Pu ] = 2 andthe average BS power is computed. The results for uniform
and Gaussian user distributions are shown in Figure 5. For
a uniform user distribution in Figure 5(a), average linkage
and complete linkage clustering provide a solution close to
the optimal set generated from the site placement and site
selection algorithms. Intuitively, since each MS is equidistant
from its neighbors, the clusters will be almost equal in size and
their centroids approximate a uniform distribution. However,
for a Gaussian user distribution in Figure 5(b), hierarchical
clustering experiences bad performance. Complete linkage and
average linkage clustering can only generate feasible network
configurations of sizes at least 72 and 78 BSs, respectively,
in comparison to 39 BSs for our proposed algorithm. Addi-
tionally, the feasible configurations incur significantly higher
power consumption. This is explained by the fact that these
linkage criteria do not take into account the discrepancy in theSIR levels and power allocations when the user density is not
constant. Generally, these results demonstrate the effectiveness
of the site selection and site placement algorithms, particularly
for non-uniform user distributions.
While decreasing the number of BSs is desirable, it makes
the uplink power requirement higher, thus increasing the
chance of crossing the Pmaxu threshold. We demonstrate theimprovement in the uplink outage due to the combined objec-
tive minimization in Figure 6(a). The outage for each network
size is computed by solving the joint planning problem for
= 0 (single objective) and = 0.5 (weighted objective)with Pmaxu = 1 W. Results demonstrate that accounting for
uplink outage in the objective minimizes the number of userscrossing the Pmaxu threshold.
D. Concurrent Voice and Data Services
We consider generalizing the model to accommodate for
concurrent voice and data services. A given percentage of
data users in the network is selected, and these data users are
independently and randomly picked from the set of all users.
To obtain insight into the operation of the algorithm under
multiple service rates, we assume all data users require 64Kbps in the downlink and 32 Kbps in the uplink and all voiceusers require 12.2 Kbps in both directions. The requirement for
higher data rate services naturally increases the network load
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30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9
10
Number of Base Stations N
PercentageofOutageUsers
With Uplink Outage Minimization (=0.5)Without Uplink Outage Minimization (=0)
5 10 20 30 40 50 60 70 80 90 950
10
20
30
40
50
60
Percentage of data users in the network
Percentageofo
utageforvoice/data
users
Voice users - 36 BS plan
Data users - 36 BS plan
Voice users - 50 BS plan
Data users - 50 BS plan
Data users
Voice users
Fig. 6. Left: Percentage of users in outage vs. number of base stations with and without the second component of (10) for Pmaxu = 1 W and a voice-onlyservice, Right: Percentage of users in outage versus the percentage of data users for two network case studies: (a) Nmin = 36 base stations, (b) Nmin = 50base stations.
and makes the QoS requirements more stringent. In Figure
6(b), we quantify this increase by considering the percentage
of voice/data users in outage versus the percentage of data
users in the network for two scenarios. The first scenario
corresponds to the optimal plan for a uniform user distribution
with 36 BSs. Since this plan is optimal in terms of power
consumption for a voice-only deployment, we expect that
adding data users will adversely impact the performance of
the network. When the network is dominated by voice users,
the data users experience 19% outage and the voice users
experience 3% outage. The rate of increase of outage as the
percentage of data users increase is relatively fast for both
traffic classes. Since this is an extreme case, we notice that
we can operate the network with a larger number of BSs
to provide more robustness to the network under concurrentvoice/data traffic. Thus, in the second scenario, we relax
Nmin to 50 BSs and we run the site placement algorithm tooptimize the site locations subject to the uniform voice/data
user distribution. It can be seen that outage drops significantly
and QoS requirements become less stringent to satisfy.
Finally, for the downlink, data traffic produces an increase
in the BS power so that all data users can also satisfy their QoS
requirements. Thus, the limiting factor becomes the total BS
power. Considering Pdmax = 30W in the simulations, we findthat a maximum of 15.5% of data users can be supported in the
first scenario whereas a maximum of 20% can be supported
in the second scenario. This analysis captures the interplay
between the traffic classes, the quality of service requirements,
and the size of the network plan.
V I. RADIO NETWORK PLANNING WITH LOCATION
CONSTRAINTS
In realistic radio network planning scenarios, operators often
do not have the liberty of placing BSs in the entire area of
the network for two main reasons. First, the presence of rough
terrains, private property such as universities, or city zoning
restrictions in the network region limits the scope of placing
BS sites. Second, the presence of radiation-sensitive zones
such as hospitals or schools places limitations on EM radiation
in such areas of the network. These limitations are referred
to in this paper as location constraints. The EM exposure
limitations are defined by specifying maximum permissible
exposure (MPE) levels for different frequency ranges [17],
[18]. In general, for frequencies lower than 300 MHz, exposure
limits are specified in terms of electric field strength (V/m) and
magnetic field strength (A/m), while for frequencies higher
than 300 MHz, exposure limits are specified in terms of power
density S (mW/cm2). Since UMTS operates at 1800 MHz,we will define our constraints in terms of power density. The
power density decays quadratically with the distance from the
BS in a free space environment. In general, measured data
shows that average power densities are generally in the 0.001 -
0.01 W/cm2 range. Health recommendations suggest that the
median exposure in urban areas be limited to 0.005 W/cm2
and that 95% of the urban population be exposed to less than
0.1 W/cm2 [17], [18].
A. Problem Formulation
In this section, we extend the formulated radio network
planning problem in Section III-A to include constraints on:
(1) Locations of deployed BSs in a target BS-free area, and
(2) peak power density in a target radiation-sensitive zone. We
model the BS-free region as a rectangular area which can be
formulated as follows:
xi / [xL,min, xL,max], yi / [yL,min, yL,max] i = 1, , N(25)
where [xL,min, xL,max] and [yL,min, yL,max] represent the con-tinuous interval of points in the BS-free region.
To include the power density constraint in the formulation,
we discretize the target area into a sufficiently representative
set of sample points. Then, we calculate an estimate of the
power received at each of the points from each BS using the
propagation model. The total power received at each point is
calculated by summing the power received from all BSs. Based
on the received power, we calculate the power density at all
points, and the sample point with the peak power density is
considered. When this constraint is satisfied, implicitly, all the
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area will satisfy the constraint. The constraint is formulated
as follows:
maxsj
Ni=1
kCi
Pdk,i
1
keq
di,sj
4f2
c2G
Sth
j = 1, , p (26)
where sj is the sample point j in the target area with j between1 and p and p is the number of sample points taken in thetarget area, f is the carrier frequency (assumed 1800 MHz forUMTS), c is the speed of light, G is the gain of the transmittingantenna of the BS in the direction of the radiation, Sth isthe recommended power density threshold, and keq and areparameters depending on the pathloss model.
The site placement problem with location constraints can
then be formulated as follows:
minx,y
U
k=1Pdk,b(k) +
U
k=1(Puk P
umax)
+ (27)
subject to (25), (26),
(11), (12), (13), (14), (15), (16), (17), (18), (19), (20)
Similarly, the site selection problem with location con-
straints can also be formulated as an extension to the initial
site selection problem as follows:
minc
N0i=1
ci (28)
s.t.
min
U
k=1
cb(k) Pdk,b(k) +
U
k=1
cb(k)(Puk P
umax)
+ (29)
subject to (25), (26),
(11), (12), (13), (14), (15), (16), (17), (18), (19), (20)
i = 1, , N0; k = 1, , U (30)
In both problems, (25) and (26) represent the location
constraints. Note that the constraints (11)-(20) from the initial
problems are also included in the formulation. The next section
describes how to extend the algorithms developed in Section
IV to solve the radio network planning problem with location
constraints.
B. Algorithm for the Radio Network Planning Problem withLocation Constraints
The location constraints as defined in (25) and (26) are
nonlinear inequality constraints. To satisfy such constraints,
we modify the algorithms presented in Section IV. A robust
extension of pattern search algorithms for general constraints
is the globally convergent ALPS which solves general prob-
lems.
Initially, the problem is modified to convert the inequality
constraints into equality constraints by introducing nonneg-
ative slack variables. Next, we attempt to find Lagrange
multiplier estimates for the equality constraints updated at each
iteration, such that the estimates do not involve information
about derivatives of the objective or constraints to be consistent
with the derivative-free nature of pattern search algorithms.
The algorithm begins with an initial value for the penalty
parameter and the Lagrange multiplier estimates. The mathe-
matical significance of these parameters is explained in [24]. A
subproblem is formulated by combining the objective function
and the nonlinear constraint function using the Lagrangian
multiplier estimates and the penalty parameters. A sequenceof such optimization problems are approximately minimized
using a pattern search algorithm such that the linear constraints
and bounds are satisfied. When the subproblem is minimized
to a required accuracy and satisfies feasibility conditions, the
Lagrangian multiplier estimates are updated. Otherwise, the
penalty parameter is increased. These steps are repeated until
the stopping criteria are met. A frequently used Lagrangian
update is the first order Hestenes-Powell multiplier update for
the augmented Lagrangian which assumes no knowledge of
derivative information [24].
Convergence analysis for the ALPS algorithm can be found
in [25], [24]. It is shown that despite the absence of any
explicit estimation of any derivatives, the pattern search
augmented Lagrangian approach exhibits first-order global
convergence properties. Although the subproblems are solved
approximately, and the stopping criterion of the subproblem is
based on the magnitude of a measure of first-order stationarity,
the algorithm converges to Karush-Kuhn-Tucker points of the
original problem. This important result establishes that pro-
ceeding by successive, inexact minimization of the augmented
Lagrangian via pattern search methods ensures convergence
[25], [24].
Since our nonlinear constraints are written in terms of
the powers allocated to users, we need to solve the set of
equations
Gd
Pd
=
2
and [Gu
] [ Pu
] =
2
for a given x (BS locations) to find the uplink and downlinkpowers, compute the value of the constraints, and solve the
inner augmented Lagrangian subproblem. Algorithm 1 and
Algorithm 2 can be extended to ALPS by constructing the
subproblem at each iteration, minimizing this subproblem over
the mesh points instead of minimizing the objective fm(x),updating the Lagrange multipliers at each successful iteration
using the Hestenes-Powell multiplier update, and adjusting the
penalty factor based on the success or failure of the polls.
C. Analysis of Network Topologies with Location Constraints
In order to demonstrate the performance of the proposed
algorithm to solve the network planning problem with loca-
tion constraints, we run the modified site selection and site
placement algorithms presented in Section VI-B consecutively
to select the smallest set of BSs that cover the network area
such that the location constraints and the QoS constraints are
satisfied. To ensure limited EM radiation within the area of
interest, we set the target peak power density in the area to
Sth=0.005 W/cm2. For computing the power density, we
assume G = 1 and f =1800 MHz. All other parameters are thesame as those listed in Table I. Figure 7 shows a particular case
with a uniform user distribution where the radiation-limited
area is a square with 4 Km x 4 Km dimensions defined as
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0 2 Km 4 Km 6 Km 8 Km 10 Km0
2 Km
4 Km
6 Km
8 Km
10 Km
Power Density (log10(W/m2))
11.5
11
10.5
10
9.5
9
8.5
8
7.5
7
3 Km 4 Km 5 Km 6 Km 7 Km3 Km
4 Km
5 Km
6 Km
7 Km
Power Density (log10(W/m2))
11
10.5
10
9.5
9
Fig. 7. A uniform user distribution with location constraints in a 4 Km x 4 Km area; Left: total area, Right: radiation-limited area. The color scale denotesthe base-10 logarithm of the power density, e.g., -9 corresponds to S = 109 W/m2.
0 500 1000 1500 2000 2500 3000 3500 400017
18
19
20
21
22
23
24
25
26
Radiation-limited area dimensions (m)
AverageBSpower
1 N
U k
=1
Pd k
,b(k)
Without Maximum Power Constraint
With Maximum Power Constraint
0 500 1000 1500 2000 2500 3000
14
16
18
20
22
24
26
Radiation-limited area dimensions (m)
AverageBSpower
1 N
k=1
Pd k,b(k)
Without Maximum Power Constraint
With Maximum Power Constraint
Fig. 8. Pareto-Optimal set that provides the tradeoff between average BS power and the dimensions of the radiation-limited area; Left: Uniform user distribution,Right: Gaussian user distribution.
follows: 3000 m x 7000 m, 3000 m y 7000 m.The figure shows that the peak power density in the area is
S = 108.5W/m2 = 3.16109W/m2 = 0.003 W/m2
which is below the predefined threshold Sth. Additionally, itis clear that all BSs satisfy as well the location constraints.
Finally, we generate pareto-optimal sets to show the tradeoff
between the dimensions of the radiation-limited area and the
achieved average BS power in the whole network. The larger
the area, the more power is required to achieve coverage to
users in the area and satisfy their SIR requirements, which
eventually increases the average BS power. Because the mod-
ified site selection and site placement algorithms are bothexecuted, the case with zero area dimensions is equivalent to
the point of operation on Figures 5(a) and 5(b) with the lowest
number of BSs and highest average BS power. Figure 8 shows
the achievable average BS powers for sample uniform and
Gaussian user distributions. For the uniform case, the square
is characterized by its center at [5000, 5000]. For the Gaussiancase, the square is characterized by: 1) yL,max touches theupper boundary, 2) The centroids of xL,min and xL,max lie atx = 5000.
We can see that with a Gaussian user distribution the slope
of increase is faster, which is why we cannot reach the 4 Km
dimension as in the uniform case; otherwise, the increase in BS
powers would prohibit satisfying the SIR requirements. The
algorithm is executed with and without a maximum BS power
constraint. The maximum BS power constraint corresponds
to (19) with Pdmax = 30 W. Such constraint will furtherlimit the area dimensions, because even though the average
BS power is lower than the constraint, some BSs (specifically
those covering the radiation-limited area) will require a very
high power to achieve the required SIR for most users. The
area dimensions in the unconstrained power case are limited
by interference whereby further increasing BS power for
some user makes the SIR requirement for other users non-
achievable. On the other hand, the area dimensions in theconstrained power case are limited by the physical maximum
BS power limit which turns out to be a more stringent
limitation. Obviously, these two sets represent specific case
studies for the area locations we considered. In fact, the slope
of the curve depends greatly on the location of the square,
especially in the Gaussian distribution case.
Another important observation is that placing a maximum
BS power constraint decreases the average BS power for some
fixed area dimensions. This is explained by the fact that when
BSs are not allowed to exceed a certain peak transmit power,
the site selection algorithm will be forced to select a larger
number of BSs to cover the entire network, specifically to
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cover users inside the constrained area. Looking back at Figure
5, we recall that the average BS power decreases as the number
of BSs increases for a given user distribution which explains
the gap between the two curves in each of Figure 8(a) and
Figure 8(b).
VII. CONCLUSION
We presented optimization-based formulations for the prob-
lems of joint uplink/downlink site placement and site selection
in cellular networks. The formulations use an SIR-based power
control mechanism with outage conditions to provide quality
guarantees. We proposed algorithms to solve the continuous
component of the problem using derivative-free optimization
techniques with general constraints. We also developed an
optimization algorithm for solving the integer component
of the problem based on a nested approach with an outer
problem that minimizes the number of base stations and an
inner problem that minimizes a cost function of the network
deployment. Case studies were presented and analyzed for
uniform and non-uniform user distributions and pareto-optimalsets were generated to tradeoff network configuration param-
eters. Finally, the formulation and solution were extended
to provide a framework for solving general radio network
planning problems with location constraints.
REFERENCES
[1] Blaustein Matha, Radio propagation in cellular networks. Artech House,Boston-London, November 1999.
[2] J. Laiho, A. Wacker, and T. Novosad, Radio network planning andoptimisation for UMTS. John Wiley & Sons, 2006.
[3] A. Mishra, Fundamentals of cellular network planning and optimisation:2G / 2.5G / 3G... Evolution to 4G. Wiley-Interscience, 2004.
[4] E. Amaldi, A. Capone, and F. Malucelli, Improved models and algo-rithms for UMTS radio planning, In Proceedings of IEEE VehicularTechnology Conference, October 2001.
[5] E. Amaldi, A. Capone, and F. Malucelli, Optimizing base station sitingin UMTS networks, In Proceedings of IEEE Vehicular TechnologyConference, October 2001.
[6] Amaldi, E. and Capone, A. and Malucelli, F. and Signori, F., Opti-mization models and algorithms for downlink UMTS radio planning,
In Proceedings of IEEE Wireless Communications and Networking
Conference (WCNC), March 2003.
[7] E. Amaldi, A. Capone, and Malucelli, Planning UMTS base stationlocation: Optimization models with power control and algorithms, IEEETransactions on Wireless Communications, vol. 2, no. 5, pp. 939952,September 2003.
[8] A. Eisenblatter, A. Fugenschuh, T. Koch, A. M. C. A. Koster, A. Martin,T. Pfender, O. Wegel, and R. Wessaly, Modelling feasible network
configurations for UMTS, Technical Report ZR-02-16, Konrad-Zuse-Zentrum fur Informationstechnik Berlin (ZIB), Germany, March 2002.
[9] E. Amaldi, A. Capone, and F. Malucelli, Radio planning and coverageoptimization of 3G cellular networks, Journal of Wireless Networks,vol. 14, no. 4, pp. 435447, August 2008.
[10] A. Eisenblatter, A. Fugenschuh, H. Geerdes, D. Junglas, T. Koch, andA. Martin, Integer programming methods for UMTS radio networkplanning, In Proceedings of WiOpt, March 2004.
[11] A. Eisenblatter and H. Geerd, Wireless network design: solution-oriented modeling and mathematical optimization, IEEE Wireless Com-munications Magazine, vol. 13, no. 6, pp. 814, December 2006.
[12] P. Calegari, F. Guidec, P. Kuonen, B. Chamaret, S. Ubeda, S. Josselin,D. Wagner, and M. Pizarosso, Radio network planning with combi-natorial optimization algorithms, In Proceedings of the ACTS MobileTelecommunications Summit, September 1996.
[13] P. Calegari, F. Guidec, P. Kuonen, and D. Wagner, Genetic approachto radio network optimization for mobile systems, In Proceedings of
IEEE 47th Vehicular Technology Conference, May 1997.
[14] J. K. J. Kalvanes and E. Olinick, Base station location and serviceassignments in WCDMA networks, INFORMS Journal on Computing,vol. 18, no. 3, pp. 366376, March 2006.
[15] J. Osepchuk and R. Petersen, Safety standards for exposure to RFelectromagnetic fields, IEEE Microwave Magazine, vol. 2, no. 2,pp. 5769, June 2001.
[16] L. Catarinucci, P. Palazzari, and L. Tarricone, Human exposure tothe near field of radiobase antennas-a full-wave solution using parallelFDTD, IEEE Transactions on Microwave Theory, vol. 51, no. 3,pp. 935940, March 2003.
[17] IEEE Standard C95. 1, IEEE standard for safety levels with respect tohuman exposure to radio frequency electromagnetic fields, 3 kHz to 300GHz, 1999.
[18] International Commission on Non-Ionizing Radiation Protection (IC-NIRP), Guidelines for limiting exposure to time-varying electric, mag-netic, and electromagnetic fields, Health Physics, vol. 74, pp. 494522,April 1998.
[19] B. Di Chiara, M. Nonato, M. Strappini, L. Tarricone, and M. Zappatore,Hybrid metaheuristic methods in parallel environments for 3G networkplanning, In Proceedings of EMC Europe Workshop, September 2005.
[20] T. Crainic, B. Di Chiara, M. Nonato, and L. Tarricone, Tackling elec-trosmog in completely configured 3G networks by parallel cooperativemeta-heuristics, IEEE Wireless Communications Magazine, vol. 13,no. 6, pp. 3441, December 2006.
[21] H. Holma and A. Toskala, WCDMA for UMTS. England: Wiley, 2000.[22] E. Dolan, R. Lewis, and V. Torczon, On the local convergence of pattern
search, SIAM Journal on Optimization, vol. 14, no. 2, pp. 567583,February 2003.
[23] R. Lewis, V. Torczon, and M. Trosset, Why pattern search works,Optima, vol. 59, pp. 17, December 1998.
[24] R. Lewis and V. Torczon, A globally convergent augmented Lagrangianpattern search algorithm for optimization with general constraints andsimple bounds, SIAM Journal on Optimization, vol. 12, no. 4, pp. 10751089, July 2002.
[25] A. Conn, N. Gould, and P. Toint, Globally convergent augmentedlagrangian algorithm for optimization with general constraints andsimple bounds, SIAM Journal on Numerical Analysis, vol. 28, no. 2,pp. 545 572, April 1991.
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